Chapter 9: Trigonometric Identities and Equations

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Chapter 9 Trigonometric Identities and
Equations

• 9.1 Trigonometric Identities

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9.1 Fundamental Identities

• Reciprocal Identities
• Quotient Identities
• Pythagorean Identities
• Negative-Number Identities
• Note It will be necessary to recognize
  alternative forms of the identities
• above, such as sin² ? 1 cos² ? and cos² ?
  1 sin² ?.

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9.1 Looking Ahead to Calculus

• Work with identities to simplify an expression
  with the appropriate trigonometric substitutions.
• For example, if x 3 tan ?, then

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9.1 Expressing One Function in Terms of Another

• Example Express cos x in terms of tan x.
• Solution Since sec x is related to both tan x and
  cos x
• by identities, start with tan² x 1 sec² x.
• Choose or sign, depending on the quadrant of
  x.

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9.1 Rewriting an Expression in Terms of Sine and
Cosine

• Example Write tan ? cot ? in terms of sin ?
  and
• cos ?.
• Solution

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9.1 Verifying Identities

• Learn the fundamental identities.
• Try to rewrite the more complicated side of the
  equation so that it is identical to the simpler
  side.
• It is often helpful to express all functions in
  terms of sine and cosine and then simplify the
  result.
• Usually, any factoring or indicated algebraic
  operations should be performed. For example,
• As you select substitutions, keep in mind the
  side you are not changing, because it represents
  your goal.
• If an expression contains 1 sin x, multiplying
  both numerator and denominator by 1 sin x would
  give 1 sin² x, which could be replaced with
  cos² x.

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9.1 Verifying an Identity ( Working with One
Side)

• Example Verify that the following equation is an
  identity.
• cot s 1 csc s(cos s sin s)
• Analytic Solution Since the side on the right
  is more
• complicated, we work with it.

Original identity
Distributive property
The given equation is an identity because the
left side equals the right side.
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9.1 Verifying an Identity

• Example Verify that the following equation is an
  identity.
• Solution

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9.1 Verifying an Identity ( Working with Both
Sides)

• Example Verify that the following equation is an
  identity.
• Solution

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9.1 Verifying an Identity ( Working with Both
Sides)

• Now work on the right side of the original
  equation.
• We have shown that